Demai, Chapter 1, Mishnah 3. This mishnah lists a series of cases that are exempt from demai.
One who buys from an am ha'aretz not for the purpose of eating:
For planting - one who buys produce from an am ha'aretz in order to sow it.
For an animal - one who buys in order to feed his animal.
Flour for hides - flour used in the process of tanning hides.
Oil for a lamp - oil bought for the purpose of kindling.
Oil for rubbing utensils - olive oil bought in order to smear and rub down one's utensils (as was the practice of oiling leather vessels to preserve them).
All of these are exempt from demai. The reason: with definite tevel we would indeed be obligated in all these things, but not by Torah law, rather by the words of the Sages, rabbinically. And since even with definite tevel the obligation is only rabbinic, in the case of demai the Sages did not require separating at all.
The Gemara in Chullin notes regarding one who buys in order to feed his animal that the exemption applies specifically when the purchase was made for this purpose from the outset. If it was bought for human consumption and afterwards he decided to feed it to his animal, he is obligated to separate demai. The same is true of every item in our mishnah: if they were bought for eating purposes and only afterwards it was decided to use them for rubbing utensils and the like, the obligation stands. The exemption applies only when the purchase from the am ha'aretz was made for these purposes from the outset.
A geographic exemption - Keziv:
A further exemption is an exemption of place: the city of Keziv, known in the book of Yehoshua as Achziv, is a city in the north, and the entire region is exempt from demai. The reason: although this region was conquered as part of Eretz Yisrael in the first conquest, in the subsequent conquest, when they came up from Babylonia, they did not conquer it and did not include it within the borders of Eretz Yisrael, and therefore it is exempt from demai. The early commentators point out that in truth even definite tevel is exempt in a region such as Keziv, only that our mishnah is dealing with demai, and therefore the matter is mentioned in the context of demai.
Additional exemptions:
The challah of an am ha'aretz - the challah that an am ha'aretz separates from his dough; we need not be concerned that it is demai and separate from it.
Terumah that became mixed in - if an am ha'aretz had definite Terumah that became mixed into his grain, and he treats it as ordinary produce, and by the rule we should have treated it as demai, we are permitted to treat it as ordinary produce and need not be concerned about demai.
"Vehalakuach bekesef ma'aser sheni" - food items purchased in Jerusalem with second tithe money. This ordinary food that the person received - we rely on the am ha'aretz having separated tithes from it properly, since he intends to sell it to someone who buys it with second tithe money.
"Ushirei hamenachos" - the remainder left from the meal offering after the handful, a full fistful, is taken and offered up on the altar.
Two reasons are given in the Yerushalmi for this exemption:
Some say that the Sages simply did not extend the decree of demai to these items.
And some explain that each one of these items carries a certain level of sanctity, and the awe of the sacred - the fear of holiness - compels the am ha'aretz to be careful and make sure that he separated tithes properly. Whether with his challah, or with something that became mixed with Terumah, or with something bought with second tithe money, or with something coming from the meal offerings in the Temple - with all of these the am ha'aretz would be careful to separate tithes.
Shemen arev:
Shemen arev is a particular type of fragrant oil containing various perfumes and spices. "Beis Shammai mechayvin" - in their view it is obligated in demai. "U'Beis Hillel potrin" - since it is used specifically for rubbing on the body alone and not for food, Beis Hillel hold that it is exempt.
To summarize: This mishnah listed the cases exempt from demai: one who buys from an am ha'aretz for planting, for an animal, flour for hides, oil for a lamp and oil for rubbing utensils - since even with definite tevel the obligation is only rabbinic, and provided that the purchase was made for this purpose from the outset; the geographic exemption from Keziv and onward, which was not conquered in the conquest of those who came up from Babylonia; the challah of an am ha'aretz, Terumah that became mixed in, what is purchased with second tithe money and the remainder of the meal offerings - whether because no decree was made in their case, or because the awe of the sacred brings the am ha'aretz to be careful with them; and the dispute between Beis Shammai and Beis Hillel regarding shemen arev, which is designated for rubbing and not for eating.